Existence of positive solutions for singular fourth-order three-point boundary value problems

In this article, we consider the boundary value problem u(4)(t)+f(t,u(t))=0, 0<t<1, subject to the boundary conditions u(0)=u′(0)=u″(0)=0 and u″(1)−αu″(η)=λ. In this setting, 0<η<1 and α∈[0,1η) are constants and λ∈[0,+∞) is a parameter. By imposing a sufficient structure on the nonlinearity f(t,u), we deduce the existence of at least one positive solution to the problem. The novelty in our setting lies in the fact that f(t,u) may be singular at t=0 and t=1. Our results here are achieved by making use of the Krasnosel’skii fixed point theorem. We conclude with examples illustrating our results and the improvements that they present.